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AFAIK, a vector can be specified using either "ordered set notation" or "matrix notation"

Ordered set notation

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Matrix notation of row and colum vectors

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I wonder if a column vector can be specified using ordered set notation. For example, can a column vector

$$ \begin{bmatrix} 1\\ 2\\ 3 \end{bmatrix} $$ be specified as follows? $$(1,2,3)$$

Plus, is the following statement correct?

A set {(1,0,0), (1,1,0), (1,0,1)} is a basis of column space of the matrix

\begin{bmatrix} 1&2&1&1\\ 0&0&1&0\\ 0&0&0&1 \end{bmatrix}.

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  • $\begingroup$ Yep. Why do you think not? $\endgroup$ – Aniruddh Agarwal Apr 29 '20 at 3:20
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    $\begingroup$ Some write $(1\;2 \;3)^T$ for $\pmatrix{1\\2\\3}$ $\endgroup$ – J. W. Tanner Apr 29 '20 at 3:23
  • $\begingroup$ In the context of vectors & matrices, I always interpret $(1,2,3)$ as a column vector. It's a way to write a column vector without leaving lots of white space on the page. $\endgroup$ – Gerry Myerson Apr 29 '20 at 3:27
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    $\begingroup$ My TA said I should've specified the basis vectors as $${{(1,0,0)^T,(1,1,0)^T,(1,0,1)^T}}$$, which I think is non-sense because it is a mix of two notation methods. $\endgroup$ – hskim Apr 29 '20 at 3:29
  • $\begingroup$ @Gerry_Myerson But how do you write row vectors, then? $\endgroup$ – J.-E. Pin Apr 29 '20 at 6:50
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I think that the column vector $\begin{bmatrix} 1\\ 2\\ 3 \end{bmatrix}$ and the ordered set $(1,2,3)$ represent the same element of the same linear space.

However, AFAIK, a set can't be seen as a basis minor of a matrix just because of the determinant minor definition.

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